Logunov–Lakshmi Priya–Sartori linear nodal-volume conjecture

Let $B\subset\mathbb{R}^{n}$ be a unit ball with $n\geq 3$. Let $u:4B\to\mathbb{R}$ be harmonic and satisfy $u(0)=0$, and let $N_u(0,\frac{1}{2})$ denote its doubling index at the indicated center and radius. Logunov–Lakshmi Priya–Sartori conjecture. There exists a constant $c>0$ depending only on $n$ such that

$$\mathcal{H}^{n-1}(\{u=0\}\cap 2B)\geq cN_u(0,\frac{1}{2}).$$

The source explicitly says that this conjecture remains open; it is the linear, and hence stronger, form suggested by the preceding almost-sharp lower bound with exponent $1-\varepsilon$.