The winning-strategy conjecture for Building Nim

Let $\operatorname{BN}(2n,\ell)$ denote the Building Nim game with parameters $2n$ and $\ell$, and let $\text{P1}$ and $\text{P2}$ be its two players. Winning-strategy conjecture. Player $\text{P1}$ wins $\operatorname{BN}(2n,\ell)$ whenever

$$2n>\ell+3.$$

Computer explorations suggest this criterion beyond the cases established in the preceding analysis, while the source notes that $\text{P1}$ has a winning strategy for $\operatorname{BN}(10,5)$ and other cases. A general proof is not provided.