The strengthened Zariski dense orbit conjecture for birational maps of dynamical degree one

Let $X$ be a smooth projective variety defined over an algebraically closed field $K$ of characteristic $0$, and let $\phi:X\dashrightarrow X$ be a birational self-map. Its first dynamical degree is $\lambda_1(\phi)=\lim_{n\to\infty}((\phi^n)^*\mathcal{L}\cdot\mathcal{L}^{d-1})^{1/n}$ for an ample line bundle $\mathcal{L}$ on $X$, where $d=\dim X$. A subvariety $Y\subset X$ is invariant under $\phi$ if $\phi|_Y$ induces a dominant rational self-map of $Y$. The strengthened Zariski dense orbit conjecture. If $\lambda_1(\phi)=1$, exactly one of the following holds: there exists a non-constant rational function $f:X\dashrightarrow\mathbb{P}^1$ such that $f\circ\phi=f$, or there exists a proper subvariety $Y\subset X$ containing every proper invariant subvariety $Z\subset X$. This strengthens the usual Zariski dense orbit conjecture by replacing the existence of a dense orbit with a common proper container for all proper invariant subvarieties; the source presents it as open.