Nadler–Tanaka's conjecture on the endomorphism spectrum of the point

Set $M=\Lambda=pt$ and consider $\Lag_{pt}(pt)$. Let

$$\cE=\operatorname{End}_{\Lag_{pt}(pt)}(pt)$$

be its naturally defined $E_\infty$-ring spectrum; write $\pi_0(\cE)$ for its zeroth homotopy group.

Nadler–Tanaka's endomorphism-spectrum conjecture. The spectrum $\cE$ is connective and

$$\pi_0(\cE)\simeq\mathbb Z.$$

This conjecture identifies the expected connective base spectrum governing the point case and is intended to support rational base change. The source does not state that it has been proved.