Index bounds for homeomorphisms of collapsed surfaces

Let $X$ be a collapse of a finite number of surfaces in one of the ways described above. Let $f:X\rightarrow X$ be a homeomorphism such that $\#\mathrm{Fix}(f)\leq\#\mathrm{Fix}(g)$ for every map $g$ homotopic to $f$, and let $x\in X$ be a fixed point of $f$.

Index-bound conjecture. One has

$$\mathrm{ind}(x)\leq 1,$$

and the fixed-point index of each isolated fixed point is also bounded below by a bound depending on the Euler characteristic of $X$.

The conjecture concerns fixed-point indices for homeomorphisms on spaces obtained by collapsing finitely many compact connected surfaces of non-positive Euler characteristic, with local surface or wedge-of-surfaces structure away from finitely many points. The proposed bounds would extend known index estimates for related wedge spaces, but the source gives no proof or resolution.