The FP-versus-symmetric subtraction conjecture for monotone 2SAT counting

Let $\text{\normalfont\textsc{\#mon2sat}}$ denote the counting problem that counts satisfying assignments of monotone 2SAT formulas, and let $[\text{\normalfont\textsc{\#mon2sat}}]^{\log}$ denote the class of functions computable with logarithmically many such oracle calls. Let $[\mathrm{FP}-\text{\normalfont\textsc{\#mon2sat}}]^{\log}$ denote the analogous class allowing an $\mathrm{FP}$ oracle together with logarithmically many monotone 2SAT counting calls, and let $[\text{\normalfont\textsc{\#mon2sat}}-\text{\normalfont\textsc{\#mon2sat}}]^{\log}$ denote the class obtained from logarithmically many differences of monotone 2SAT counting calls.

FP-versus-symmetric subtraction conjecture. $[\mathrm{FP}-\text{\normalfont\textsc{\#mon2sat}}]^{\log}$ is strictly contained in $[\text{\normalfont\textsc{\#mon2sat}}-\text{\normalfont\textsc{\#mon2sat}}]^{\log}$.

The paper has established that logarithmically many symmetric differences of monotone 2SAT counting calls capture the relevant hardness of $\mathrm{gap}\mathrm{P}$, while it leaves open whether one call can be replaced by an $\mathrm{FP}$ oracle. The authors expect that simulating negation by subtraction requires symmetry, which motivates the strict-containment conjecture.