The safety-bound conjecture for Segre varieties

Let $X=\mathbb{P}^{n_1}\times\dots\times\mathbb{P}^{n_k}$ be a Segre variety with $n_1\leq\dots\leq n_k$. Let $\mathscr{O}_-$ and $\mathscr{O}_+$ be the safety bounds of $X$, and let $T(n_1,\dots,n_k;n_k;0,\dots,0,a_k)$ be the statement of room $\mathscr{R}$ appearing below.

Safety-bound conjecture. The following statements hold:

• The two safety bounds are symmetric:

$$\mathscr{O}_-=-\mathscr{O}_+.$$

• The positive safety bound is

$$\mathscr{O}_+=\left(\sum_{i=1}^{k-1}n_i-1\right)n_k.$$

• If

$$a_k=\prod_{i=1}^{k-1}(n_i+1)-(n_k+1)\geq0,$$

then the statement $T(n_1,\dots,n_k;n_k;0,\dots,0,a_k)$ has room $\mathscr{R}=\mathscr{O}_-$ and is false.

The conjecture is suggested by explicit computations of low-dimensional Segre varieties and by the observed agreement of the positive and negative safety bounds with the stated formula. Its general validity, including the final assertion about the statement $T$, remains open.