Let $N$ and $N'$ be regular neighborhoods of a subpolyhedron $P$ in a closed PL manifold $M$, and suppose that $t$ is a free PL involution on $M$ such that each of $\partial N$, $\partial N'$ is invariant under $t$. Does there exist an equivariant PL isotopy of $M$ taking $N$ onto $N'$?

Edit: Beware that $N$ is not assumed to be $t$-invariant. The case where $N$ is $t$-invariant is of course trivial.

(Example: $M=S^m$ with $t$ the antipodal involution, $P$ is a point, $N$ is a hemisphere and $N'$ something fancier.)

I also do not know the answer to the similar question in the smooth category, where $N$ and $N'$ are tubular neighborhoods of a submanifold.

It does not seem like usual arguments about regular neighborhoods work to prove this, but also I don't immediately see a counterexample.

(As a side remark, I'm ultimately interested in relative regular neighborhoods and a non-free involution whose fixed point set is forced to lie in $\partial N$ by the choice of the relativization.)