Right this minute, certainly not. They are freaked out and don't even know Rand shares LTT's memories.

But things may well change between the AS and Rand before the LB is over. Things have to evolve between Rand and Egwene at some point. Elayne is Aes Sedai, he has Nynaeve on his side, and Cadsuane and soon he will have Moiraine, who is bound to throw her weight on Rand's side (and who may well give the Tower the same sort of brutal wake-up call she gave the Emond's Fielders with her speech on how low they had fallen since the days of Manetheren). I just can't see how a Tower lead by Egwene with Siuan as one of her advisors that would be on the verge of a break-up with Rand on the eve of the LB could ever sit well with Moiraine. If that's the way it's going, Moiraine will do all she can to prevent this.

Aviendha isn't AS though close friend to the Amyrin and Elayne, and she has a skill related to OP objects. Elayne is massively interested in OP objects and taking risks studying them and making copies. Elayne and Rand will be together once more very soon. Eventually Rand will know of interest and talents. Why would Rand refrain from sharing some of his knowledge with Elayne? Aside from Elayne's collection (which could well be seized soon by the Shadow... there's in there a object that seems likely to be an extra 'Talisman of Growing" with which it's possible to reopen the Waygates closed by Loial... I'm not quite sure it will be the Light using that object in the end...), there's a stockpile from Rhuidean left warded by Rand in Cairhien, and the vaults in Tear warded by Rand as well. When the useful objects from those vaults are brought into play during TG, why would the Tower stupidly refuse Rand's help to identify the objects from their own collection which could be useful and learn from him how they're used? Liandrin's group got a few nasty things from there, there's bound to be more useful stuff left (all the more since on the list of objects they were ordered to steal, Liandrin had only small things easy to carry off). If the WT objects, there's Nyaneve, Cadsuane, Elayne, Moiraine as well as Cadsuane's circle of followers to jump into the debate... it's all the more important that several objects are probably meant for men.

A few key Aes Sedai in Rand's inner circle are already aware that Rand has knowledge of OP objects. Recently he did tell Cadsuane the name and real purpose of her apparatus, remember. The matter with the dreamspike is not over either as there's a second one protecting the BT. I seriously doubt Rand won't be involved, and unlike Perrin and Egwene he knows exactly how this device works and what it does. Plain evidence that he has AOL knowledge and skills is bound to surface, and sooner or later Egwene will have to accept it. It wouldn't be difficult for Rand to convince Egwene and the WO, Rand merely has to talk to them of TAR issues they don't know about.

There may be yet another big clash between the AS and Rand coming soon, but it's pretty obvious Jordan also intended to unite the Light before it's over. The obstacles to that have fallen one after another, slowly. The big one that remains is the matter of the seals between Rand and Egwene.

You may roll your eyes all you want, but the possibility Rand contributes to putting into play OP objects from the WT vaults during TG is very real.