Conjecture for $d=1$: Set $p:=[\frac{n}{2}]$ (integral part). Define the sequence $v_1,\ldots,v_p$ 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 $$a^2\sum_1^pv_j^2=\frac12.$$ It corresponds to the projection on the line spanned by $u:=(-a_1,\ldots,-a_p, (0,) \, a_p,\ldots,a_1)$ where $a_j:=av_j$. The zero is present if $n$ is odd.
In this construction, the equality $\|Pe_I\|=\|e_I\|$ ($I$ a subset of indices) is achieved for every subset $I$ of $n-p+1,\ldots,n$.
Asymptotically, we have $v_k\sim c k^{-1/2}$ and $a\sim c'(\log n)^{-1/2}$.