The co-chromaticity conjecture for signed complete graphs

A signed graph is a graph whose edges are assigned positive or negative signs; its bivariate chromatic polynomial is denoted by $\mathscr B(\Sigma,x,y)$, and its chromatic polynomial by $\mathsf E(\Sigma,x,y)$. Two signed graphs are co-chromatic when they have the same chromatic polynomial. A signed complete graph is a signed graph whose underlying graph is complete. Co-chromaticity conjecture. There are no co-chromatic signed complete graphs. This asserts that signed complete graphs cannot share the same chromatic polynomial unless they are the same signed graph in the relevant sense. The statement is presented as a conjecture, and no resolution is supplied in the source.