Conjecture on identifiability of binary Segre products

Let $m>5$, and let $k$ be an integer satisfying

$$1\leq k\leq \left\lfloor \frac{2^m}{m+1}\right\rfloor-1.$$

The binary Segre product is the variety $({\mathbb{P}}^1)^m$.

Identifiability conjecture. For all such $m$ and $k$, the binary Segre product $({\mathbb{P}}^1)^m$ is $k$-identifiable.

This conjecture proposes identifiability throughout the full range allowed by the elementary dimensional bound. The paper proves the weaker bound $k+1\leq 2^{m-1}/m$ and shows that $({\mathbb{P}}^1)^5$ is not $4$-identifiable; the conjectured statement remains open in the indicated generality.