I think that the answer is "yes" if by TQFT you mean one that extends all the way down to 1-manifolds. The MTC is the thing assigned to a circle by this extended TQFT.

There's also lurking somewhere in here the issue of whether you mean 3d TQFT "with anomaly." The 3d TQFTs coming from modular tensor categories via RT really have a little bit of 4d structure.

I think that the answer is "yes" if by TQFT you mean one that extends all the way down to 1-manifolds. The MTC is the thing assigned to a circle by this extended TQFT.

There's also lurking somewhere in here the issue of whether you mean 3d TQFT "with anomaly." The 3d TQFTs coming from modular tensor categories via RT really have a little bit of 4d structure.

Edit: Removed a false sentence. Also note that precise statements (and eventually a proof, once all parts are out) can be found in work of Bartlett-Douglas-Schommer-Pries-Vicary

I think that the answer is "yes" if by TQFT you mean one that extends all the way down to 1-manifolds. The MTC is the thing assigned to a circle by this extended TQFT.

Furthermore I suspect that this might be proved in Freed-Hopkins-Lurie-Teleman, but I don't actually understand that paper so I can't be sure. There's also lurking somewhere in here the issue of whether you mean 3d TQFT "with anomaly." The 3d TQFTs coming from modular tensor categories via RT really have a little bit of 4d structure.

I think that the answer is "yes" if by TQFT you mean one that extends all the way down to 1-manifolds. The MTC is the thing assigned to a circle by this extended TQFT.

There's also lurking somewhere in here the issue of whether you mean 3d TQFT "with anomaly." The 3d TQFTs coming from modular tensor categories via RT really have a little bit of 4d structure.