*Edited*. Conjecture for $d=1$: Define the sequence $v_1,\ldots,v_n$ by $v_1=1$ and 
$$v_{k+1}=\left(\sqrt{1+\frac1k}-1\right)(v_1+\cdots v_k).$$
Then the $\min\max$ equals $a$ where 
$$a^2\sum_1^nv_j^2=1.$$
It corresponds to the projection on the line spanned by $u:=(a_1,\ldots,a_n)$ where $a_j:=av_j$. 

In this construction, the equality $\|Pe_I\|=\|e_I\|$ ($I$ a subset of indices) is achieved for every subset $I=(1,\ldots,p)$ with $1\le p\le n$. 

We have $v_k=\sqrt{k}-\sqrt{k-1}\sim\frac{1}{2\sqrt k}$. Asymptotically, we have  $a\sim\frac{2}{\sqrt{\log n}}$.