Monotonicity conjecture for the stability range of the delayed logistic map

Let $K$ be the carrying capacity, $r$ the growth parameter, and $\tau$ a nonnegative integer time delay. Consider the delayed logistic recurrence

$$X_{n+1}=X_n+rX_n\Bigl(1-\frac{X_{n-\tau}}{K}\Bigr).$$

For a fixed delay, write the two fixed points as

$$X_1=(0,0,\ldots,0),\qquad X_2=(K,K,\ldots,K),$$

with $\tau+1$ coordinates. Monotonicity conjecture for the stability range. For every $\tau$, these are the only two fixed points, and $X_2$ is stable precisely when

$$0<r<f(\tau),$$

where $f:\mathbb{N}\to\mathbb{R}$ is a monotone decreasing function of $\tau$. The computed cases indicate that increasing the delay shrinks the stability range of the non-trivial fixed point, consistent with the general destabilizing effect of delay in population models; the monotonicity of the stability threshold is the asserted claim.