Coincidence point set conjecture for homotopic fixed point sets

Let $X$ be a contractible digital image and let $f:X\longrightarrow X$ be a continuous mapping. Here, $HFS(f,f)$ denotes the homotopic fixed point set for the pair $(f,f)$, $S(f)$ denotes the set of possible fixed-point cardinalities among maps homotopic to $f$, $CFS(X)$ denotes the coincidence fixed point set of $X$, and $F(X)$ denotes the set of possible fixed-point cardinalities on $X$. Coincidence point set conjecture. Then

$$HFS(f,f)=S(f)\quad\text{and}\quad CFS(X)=F(X).$$

The conjecture addresses whether the two-map fixed-point invariants reduce to their ordinary fixed-point counterparts when the maps coincide. The supplied text gives the statement in response to preceding questions and verifies the corresponding equalities for a particular contractible three-dimensional digital image, but does not establish the claim for every contractible digital image.