Non-positive surface curvature implies solvable word problem

Let $X$ be a finite, standard, irreducible 2-complex, and let $\sigma_+(X)$ denote its positive surface curvature invariant.

Non-positive surface curvature implies solvable word problem. If $\sigma_+(X)\leq 0$ then the word problem is solvable in $\pi_1(X)$.

The conjecture proposes a surface-curvature explanation for known solvable-word-problem results in CAT(0), small-cancellation and one-relator settings. The source also warns that non-positive surface curvature can coexist with very large isoperimetric inequalities; no resolution is supplied.