Set families (possibly parameterized) - Combinatorial Parameters

[ben1997online], page 54: the domain has $2.2^d+1$ elements ${z,x_1,\ldots,x_{2^d},y_1,\ldots,y_{2^d}}$ and $\mathcal{H}$ consists of $2.2^d$ functions ${f_1,\ldots,f_{2^d},g_1,\ldots,g_{2^d}}$ such that $f_i(z)=0$, $g_1(z)=1$, and $f_i(x_j)=g_i(y_j)=1_{i=j}$ and we split the elements $x_1,\ldots,x_{2^d}$ and $y_1,\ldots,y_{2^d}$ into $2^d/d$ blocks of size $d$. On each of the blocks the $2^d$ functions $f_1,\ldots,f_{2^d}$ take all $2^d$ possible combinations of values (but they are identical on each of the blocks, so $f_1$ is $0$ on all the $y_i$ and $f_{2^d}$ is equal to $1$ on all the $y_i$). We have $VC=d$, $RTD=2$ (always take $z$ and $x_i$ or $z$ and $y_i$), $M_{sd}\le 2$, but $osh_{min}=d=\Omega(\log n)$ (\olivier{proof?}) hence $M_{best}=\Omega(\log n)$. Do we have $Q_{par}=\Omega(\log n)$?