Borozenets' rank–crank inequalities modulo 11

Let $M(m,n)$ and $N(m,n)$ denote, respectively, the numbers of partitions of $n$ with crank and rank $m$. Let $M(a,c;n)$ and $N(a,c;n)$ denote, respectively, the numbers of partitions of $n$ with crank and rank congruent to $a\ \, \left( \mathrm{mod} \, c \right)$. For sequences $a=\{a(n)\}_{n=1}^{\infty}$ and $b=\{b(n)\}_{n=1}^{\infty}$, write $a\le_N b$ when $a(n)\le b(n)$ for $n\ge N$, and write $a\le b$ when $a(n)\le b(n)$ for $n\ge0$. For $0\le j,d<11$, define

$$N_{j,d}(n):=N(j,11;11n+d),\qquad M_{j,d}(n):=M(j,11;11n+d),\qquad p_d(n):=p(11n+d).$$

Borozenets' conjecture. The following chains of inequalities hold:

$$\begin{aligned} N_{5,0}&\le N_{4,0}\le_2 N_{3,0}\le M_{1,0}\le \frac{p_0}{11}\le M_{0,0}\le_1 N_{2,0}\le N_{1,0}\le_3 N_{0,0},\\ N_{5,1}&\le N_{4,1}\le N_{3,1}\le_1 M_{0,1}\le M_{2,1}\le \frac{p_1}{11}\le M_{1,1}\le_1 N_{2,1}\le N_{1,1}\le N_{0,1},\\ N_{5,2}&\le N_{4,2}\le N_{3,2}\le M_{0,2}\le \frac{p_2}{11}\le M_{2,2}\le_1 N_{2,2}\le N_{1,2}\le_3 N_{0,2},\\ N_{5,3}&\le N_{4,3}\le N_{3,3}\le M_{1,3}\le \frac{p_3}{11}\le M_{0,3}\le N_{2,3}\le_1 N_{1,3}\le_2 N_{0,3},\\ N_{5,4}&\le_3 N_{4,4}\le N_{3,4}\le_1 M_{1,4}\le \frac{p_4}{11}\le M_{0,4}\le_1 N_{2,4}\le N_{1,4}\le_3 N_{0,4},\\ N_{5,5}&\le N_{4,5}\le_1 N_{3,5}\le M_{2,5}\le \frac{p_5}{11}\le M_{0,5}\le N_{2,5}\le N_{1,5}\le N_{0,5},\\ N_{5,6}&\le_1 N_{4,6}\le N_{3,6}\le \frac{p_6}{11}\le N_{2,6}\le N_{1,6}\le_1 N_{0,6},\\ N_{5,7}&\le N_{4,7}\le N_{3,7}\le M_{0,7}\le \frac{p_7}{11}\le M_{1,7}\le N_{2,7}\le_1 N_{1,7}\le N_{0,7},\\ N_{5,8}&\le N_{4,8}\le N_{3,8}\le M_{0,8}\le_3 \frac{p_8}{11}\le_3 M_{1,8}\le N_{2,8}\le N_{1,8}\le_3 N_{0,8},\\ N_{5,9}&\le N_{4,9}\le N_{3,9}\le_1 M_{0,9}\le \frac{p_9}{11}\le M_{1,9}\le N_{2,9}\le N_{1,9}\le_2 N_{0,9},\\ N_{5,10}&\le N_{4,10}\le N_{3,10}\le_1 M_{3,10}\le \frac{p_{10}}{11}\le M_{0,10}\le N_{2,10}\le N_{1,10}\le_3 N_{0,10}. \end{aligned}$$

These inequalities were conjectured by Borozenets and are the subject of the paper's proofs; the supplied source does not explicitly provide a resolution status for this statement.