Equality of Euler excess and skewness for complete graphs of order divisible by six

Prove that for every integer m ≥ 1, every complete graph K_{6m}, and every orientable surface S_t with −1 ≤ t ≤ 2m² − 3m + 1, the t-excess δ_t(K_{6m}) equals the t-skewness μ_t(K_{6m}).

Background

Section 4 develops constructions relating Euler excess and skewness for complete graphs on orientable surfaces of higher genus. After describing a proposed handle-based construction for K_n when n is divisible by six, the paper formulates a conjecture covering a range of surface genera for K_{6m}.

The conjecture concerns the equality δ_t(K_n) = μ_t(K_n), where δ_t is the Euler-formula lower bound and μ_t is the minimum number of edges whose deletion makes the graph embeddable in S_t. The stated range includes the surfaces addressed by the proposed construction, but the claim itself is presented as unresolved.