Let $M$ be a compact contractible manifold, $X\subset\partial M$ and $C_X$ the cone over $X$.

Question: Is it true that $C_X$ embeds in $M$ with its boundary $\partial C_X$ mapped to $X\subset \partial M$?

I am mostly interestied in the piecewise linear case, that is, $M$ is a PL manifold, $X$ is a simplicial complex in $\partial M$, the embedding is a PL map, etc. I am also mostly interested in the case when $M$ is a 4-manifold, but a general answer is welcome too.

Note that the answer is "Yes" if $M$ is a ball.