Preimage-of-a-cycle conjecture for PL maps in strong general position

Let $c,n,d$ be non-negative integers such that $c<d$. Let $C$ be a $c$-chain in $d53d^d$, let $N$ be an $n$-dimensional closed PL manifold, and let $f:N\tod53d^d-\partial C$ be a PL map in strong general position with respect to $C$. Preimage-of-a-cycle conjecture. The preimage $f^{-1}C$ is the support of a $(c+n-d)$-cycle in $|T_N|\subsetd53d^m$. This asserts that transverse preimages of chains without boundary inherit a cycle structure; the supplied material does not indicate whether the claim has been proved or refuted.