The tmf Tate-spectrum non-splitting conjecture at the prime 3

Let $\mathit{tmf}$ denote the connective topological modular forms spectrum, let $\Sigma_3$ be the symmetric group on three letters, and let $\mathit{tmf}^{t\Sigma_3}$ denote its Tate spectrum. Write $BP\langle 1\rangle$ for the height-one truncated Brown–Peterson spectrum. The tmf Tate-spectrum non-splitting conjecture. There does not exist a splitting

$$\mathit{tmf}^{t\Sigma_3}\simeq\bigvee_{k=-\infty}^{\infty}\Sigma^{12k-1}BP\langle 1\rangle.$$

The preceding computation identifies the homotopy of $\mathit{tmf}^{t\Sigma_3}$ as an indecomposable $\mathit{tmf}_*$-module. The Adams filtration pattern is inconsistent with the analogous wedge decomposition, motivating this conjectural non-splitting result.