Tridiagonal pairs of $q$-Racah type and the $\mu$-conjecture

Let $\text{K}$ denote a field and let $V$ denote a vector space over $\text{K}$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$and$A^:V \to V$thatsatisfythefollowingconditions:(i)eachof$A,A^$isdiagonalizable;(ii)thereexistsanordering$\lbrace V_i\rbrace_{i=0}^d$oftheeigenspacesof$A$suchthat$A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$for$0 \leq i \leq d$,where$V_{-1}=0$and$V_{d+1}=0$;(iii)thereexistsanordering$\lbrace V^i\rbrace{i=0}^\delta$oftheeigenspacesof$A^$suchthat$A V^_i \subseteq V^{i-1} + V^*{i} + V^_{i+1}$for$0 \leq i \leq \delta$,where$V^{-1}=0$and$V^*{\delta+1}=0$;(iv)thereisnosubspace$W$of$V$suchthat$AW \subseteq W$,$A^* W \subseteq W$,$W \neq 0$,$W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=\delta$ and for $0 \leq i \leq d$thedimensionsof$V_i$,$V_{d-i}$,$V^_i$,$V^_{d-i}$ coincide. We say the pair $A,A^{\ast }$ is {\it sharp} whenever $\dim V_0=1$. It is known that if $\text{K}$ is algebraically closed then $A,A^{\ast }$ is sharp. A conjectured classification of the sharp tridiagonal pairs was recently introduced by T. Ito and the second author. Shortly afterwards we introduced a conjecture, called the {\em $\mu$-conjecture}, which implies the classification conjecture. In this paper we show that the $\mu$-conjecture holds in a special case called $q$-Racah.