Combinatorial characterization of zero mutable Laurent polynomials on rectangular triangles

For positive integers $a,b$, let $\textup{ZMLP}(a,b)$ denote the zero mutable Laurent polynomials associated with the rectangular triangle $\triangle(a,b)$, and assume $\gcd(a,b)=1$. Let $\textup{ZMLP}_{\textup{comb}}(a,b)$ be the set of pairs $(\mathbf a,\mathbf b)$ of decreasing partitions of $a$ and $b$ satisfying

$$\sum_i a_i^2+\sum_j b_j^2=ab+1,\qquad \max(\mathbf a)\leq b,\qquad \max(\mathbf b)\leq a,$$

with $\max(\mathbf a)+\max(\mathbf b)\leq\max(a,b)$.

Combinatorial characterization conjecture. If $\gcd(a,b)=1$, then

$$\textup{ZMLP}(a,b)=\textup{ZMLP}_{\textup{comb}}(a,b).$$

The combinatorial set is known to contain $\textup{ZMLP}(a,b)$, so the conjecture asserts that all pairs satisfying the necessary conditions arise from zero mutable Laurent polynomials. The supplied text does not state a general resolution.