The forcing flexibility conjecture for automorphism-tower heights

Let $V$ be the ground model. For a centreless group $G\in V$, let $\tau(G)$ be the least ordinal $\alpha$ such that the automorphism tower satisfies $G_\beta=G_\alpha$ for all $\beta>\alpha$, and let $\tau^{V^{\mathbb{P}}}(G)$ denote the value computed in the forcing extension $V^{\mathbb{P}}$. Forcing flexibility conjecture. Let $\alpha$ and $\beta$ be ordinals such that if $\alpha\geq 1$, then $\beta\geq 1$. Then there exist a centreless group $G$ and a notion of forcing $\mathbb{P}$ such that

$$\tau(G)=\alpha\quad\text{and}\quad\tau^{V^{\mathbb{P}}}(G)=\beta.$$

This conjecture asks whether the height of an automorphism tower can be changed to any prescribed ordinal height by passing to a suitable forcing extension, subject only to the fact that a nonzero height in the ground model remains nonzero in the extension. Earlier results showed that automorphism towers of centreless groups terminate and that their heights can increase or decrease in generic extensions, but the full simultaneous realization asserted here was not known from the supplied context.