The minimal decomposition conjecture for shoreless continua

Let $X$ be a continuum with no shore points, and let

$$X=X_1\oplus\cdots\oplus X_N$$

be a minimal decomposition, meaning a decomposition minimal in the ordering used in the paper. Suppose that

$$p\in X_1\cap\cdots\cap X_N.$$

The minimal decomposition conjecture. The point $p$ is non-coastal when treated as a point of each $X_n$. This asserts the desired non-coastal behavior for every minimal decomposition, while the surrounding discussion presents the broader decomposition claim as a stronger, false conjecture.