Unbounded ratio between consecutive latency domination numbers

Prove or disprove whether, for every latency L≥0, the supremum over Young diagrams and Hamming rectangles of the ratio γ^L(𝒴,m,n)/γ^{L+1}(𝒴,m,n) is infinite.

Background

Theorem 3.5 proves that the ratio between consecutive latency parameters is unbounded for L=0,1,2. The authors leave open whether the same phenomenon persists for every nonnegative latency. Resolving this would characterize how sharply the minimum initial-set size can decrease when one additional growth step is allowed.