The modal multiplicity conjecture for sieve patterns

Let $p_n$ denote the $n$th prime, and let $\mathcal{P}_{p_n}$ denote the corresponding sieve pattern. Modal multiplicity conjecture. The number $6$ occurs most frequently among the multiplicities of numbers in any pattern $\mathcal{P}_{p_n}$ when $n\geqslant 11$. This is an empirical claim about the frequency distribution of values in the sieve patterns; the supplied text gives no proof or resolution.