Conjecture: Sampford method satisfies threshold monotonicity

Prove that the apportionment method induced by Sampford sampling satisfies threshold monotonicity, i.e., for any fixed house size h, any two vote vectors v and v′ such that every party i in a coalition T has a weakly larger standard quota q_i′ = h·v_i′/(∑_j v_j′) than q_i = h·v_i/(∑_j v_j) and every party outside T has a weakly smaller standard quota, the probability that T is awarded at least s seats weakly increases for every threshold s ∈ ℕ.

Background

Having shown that Sampford sampling achieves selection monotonicity for rounding rules, the authors turn to apportionment-level monotonicity. They provide impossibility results for pairwise threshold monotonicity under full support and for vote-count-based axioms, but leave open whether threshold monotonicity (defined via vote shares/standard quotas) can be satisfied.

They specifically conjecture that the apportionment method induced by Sampford sampling meets threshold monotonicity, supported by empirical checks and its favorable correlation properties, while noting that a general proof is currently lacking.