Mixed CR minimisers for homological systoles in Berger projective spaces

Let $N\geq 2$, let $1\leq k\leq 2N-2$, set $d=k+1$, and let $\mu>0$. For the Berger metric $g_\mu$ on $\mathbb {RP}^{2N-1}$, define the mod-two homological systole by

$$\operatorname{sys}_k^{\mathbb Z_2}(g_\mu)=\inf\{\mathbf M_{g_\mu}(T):T\in\mathcal Z_k(\mathbb {RP}^{2N-1};\mathbb Z_2),\ [T]\ne0\}.$$

For a real $d$-plane $V\subset\mathbb R^{2N}\cong\mathbb C^N$, let $\mathbb {RP}(V)$ denote its associated linear projective $k$-plane. Mixed CR minimisers conjecture.

$$\operatorname{sys}_k^{\mathbb Z_2}(g_\mu)=\min_{V\in\operatorname{Gr}_d(\mathbb R^{2N})}\operatorname{Vol}_{g_\mu}(\mathbb {RP}(V)).$$

If $\mu\ne1$, every minimiser is, modulo $U(N)$,

$$V\cong\begin{cases} \mathbb C^{c_{\min}}\oplus\mathbb R^{d-2c_{\min}},&\mu>1,\\ \mathbb C^{c_{\max}}\oplus\mathbb R^{d-2c_{\max}},&0<\mu<1, \end{cases}$$

where $c_{\min}=\max\{0,d-N\}$ and $c_{\max}=\lfloor d/2\rfloor$. When $\mu=1$, every round linear projective $k$-plane is a minimiser, and the equality family is the full $O(2N)$-orbit. The conjecture concerns precisely the remaining mixed regimes: $\mu>1$ with $k>N-1$ and $0<\mu<1$ with $k$ even; the pure cases are proved by the exact homological systole theorems, while a global inequality for the mixed cases remains to be established.