A volumish theorem for alternating virtual links

Dasbach and Lin proved a "volumish theorem" for alternating links. We prove the analogue for alternating link diagrams on surfaces, which provides bounds on the hyperbolic volume of a link in a thickened surface in terms of coefficients of its reduced Krushkal-Jones polynomial. Along the way, we show that certain coefficients of the 4-variable Krushkal polynomial express the cycle rank of the reduced Tait graph on the surface.