The decomposition conjecture for shoreless continua

Let $X$ be a continuum with no shore points, let $p\in X$, and let $X=X_1\oplus\cdots\oplus X_N$ denote a decomposition into subcontinua. A point is non-coastal in a subcontinuum if it is not a coastal point there. The decomposition conjecture. There is a decomposition

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

such that

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

and $p$ is non-coastal when treated as a point of each $X_n$. The claim is intended to make the spot-welding construction of two copies of ${\mathbb H}^*$ generic, but the stronger conjecture is false.