Let $k$ be an (fixed, $3$ for instance) integer, what is the fastest simulation of a $k$ tape Turing machine by a two tape Turing machine? That is we're looking for the best 2 tape TM $U$, such that when given $<\#M,x>$ as input where $\#M$ is a representation of $M$ a $k$ tape TM, $U$ returns $M(x)$
As far as I know the best we can do is a logarithmic factor slowdown. Meaning the fastest $U(<\#M,x>)$ we know will stop in $O(T\log T)$ steps if $M(x)$ takes $T$ steps to stop. However I've read here and there in the literature and on this forum that if $k$ is fixed, then we can get rid of that logarithmic overhead. For instance this comment https://cstheory.stackexchange.com/a/33937/55419 (but it's not the only place in which I've read it).
Here is a paper first showing the $T\log T$ slowdown. There doesn't seem to be anything indicating that we may go faster if $k$ is fixed.
I thought about it for a little while and I don't see any way to have a constant factor slowdown, even if it just were simulating $3$ tapes Turing machines with $2$ tapes. It's important for a result I'm writing if we could indeed have a constant factor slowdown, if you believe it to be true could you provide a proof?