I would suspect that the problem has more to do with the uneven surface of the pebble tile trapping water than the grout itself.

Overall though, you should not have to clean sinks every 2 days to keep the mildew at bay. It sounds like you have a major moisture/ventilation problem in your home.

A couple of points to ponder...is this a localized area of increased humidity, or do you have whole-house elevated humidity?

Has the relative humidity in your house ever been checked? A whole-house dehumidifier might help, they're fairly easy to install if you have whole-house air conditioning.

If you already have one, it just might not be running as much now due to the AC not being used as much in the cooler winter months, so your winter interior levels might be increased.

The bathroom fan you have can help evacuate localized moisture, but if the overall humidity in the house is high enough for mold/mildew to flourish elsewhere in the house then it might not be a fault with the pebble. The large amount of grout that comes along with most pebble installations can be an irritation for sure if you have mold/mildew.

Do you have issues in other bathrooms or in other rooms in your house?

If so, my first recommendation from this "whole house" side of the argument would be to look into the humidity levels in your house. By the way, installing a whole house dehumidifier would probably be less expensive than redoing your shower floor.

Let's toss whole house aside and just consider the bathroom:

When your shower floor was built, was the waterproof membrane put on top of a sloped surface, or was the waterproof membrane set on the flat subfloor and the sloped surface put on top of the flat membrane?

Membranes put flat on the subfloor can hold water, exacerbating moisture problems on the shower floor. Especially a pebble floor. That excess moisture can then increase the localized humidity levels in the bathroom.

Investigate the efficiency of your bathroom fan. How big the bath is versus the rated size of the fan. Run the fan. Is there any significant air movement? Will is suck and hold a piece of toilet paper to the face of the grill? Does the airflow grab the toilet paper and snap it onto the grill or does it barely hold on to it? Is it strong air movement? Weak? I'd take a look to where the bath fan outlet is outside your house. Are there any impediments to air flow? Bird nest, bee hive, etc. The duct work from the bathroom fan to the outlet hood...is the duct crushed anywhere? Pinched as it squeezes through a framing bay? Too long of a run? Too many turns?

Grout sealers...some are film sealers, some penetrating. Film sealers are bad. They sound good, but in a bathroom it's like putting saran wrap over your tile. It's great...until moisture gets behind the wrap. Which it will. The better choice would be a penetrating sealer that is moisture vapor permeable. Those sealers will bead water droplets and will help the grout shed water. But if moisture gets under the grout (which it will) they allow the moisture vapor to permeate through, back out of the grout and back to free air within the room.

Stepping back a bit to the pebble floor itself. Does it drain well or does water pool after you're done showering? Also, since pebbles can be a bit of work to grout, some people mix the grout too wet so it flows easier. That's bad. The water that isn't consumed during the hydration process will eventually evaporate away, and instead of having nice dense grout you have grout with micro-voids within the cementitious structure. It'll sort of act like a sponge, collecting and holding water.

So...it could be myriad things. If this bathroom is the only moldy place in your house, start with the bathroom. The fan, is it moving air. The shower pan, is the membrane flat or sloped.

I'd be looking for mold issues somewhere inside the walls of your home. I'd be willing to bet there's a well established mold colony somewhere in your home. Even if the pebble tile is holding water, there has to be mold spores coming from somewhere if it keeps coming back.

Hey Catz!
I have the same shower floor with marble walls and there is nothing wrong with your shower, no vent problems, no mold issues in other places in your home, nothing. It's just that there is a ton of grout around all of those pebbles and grout is pourous and holds water. I keep a small spray bottle of undiluted pure bleach in my shower and when I see a section of grout starting to darken and mildew, I shoot it with the bleach when I get out of the shower and poof, it's gone. I love the floor and it looks great!

"I have the same shower floor with marble walls and there is nothing wrong with your shower, no vent problems, no mold issues in other places in your home, nothing. It's just that there is a ton of grout around all of those pebbles and grout is pourous and holds water. "

Couldn't agree less. The OP's problem is not contained to the shower floor, so obviously the grout isn't the only, or even main, issue.

Pebble Tile may be the least of the factors causing this mold. I vote to go hunting for bigger factors.