The Erdős–Ko–Rado conjecture for tilings with arbitrary finite tile sets

Let $\Lambda\subseteq\mathbb N$ be a finite nonempty set of allowed tile lengths. For a positive integer $n$, let $\mathcal T_n^\Lambda$ be the set of partitions of $[n]=\{1,2,\ldots,n\}$ into intervals of lengths in $\Lambda$, and let $\mathcal I(T)$ denote the set of tiles in a tiling $T$. A family $\mathcal F\subseteq\mathcal T_n^\Lambda$ is intersecting if every two distinct tilings in $\mathcal F$ have a common tile. Let $a_m^\Lambda=|\mathcal T_m^\Lambda|$, with $a_0^\Lambda=1$ and $a_m^\Lambda=0$ for $m<0$. Set

$$\lambda_0=\min\Lambda.$$

Erdős–Ko–Rado conjecture for tilings. For sufficiently large $n$, every intersecting family $\mathcal F\subseteq\mathcal T_n^\Lambda$ satisfies

$$|\mathcal F|\leq a_{n-\lambda_0}^\Lambda.$$

This would extend the proven bound for tile sets containing $1$, where the endpoint stars have size $a_{n-1}^\Lambda$, and the known $2\times n$ tilings theorem for $\Lambda=\{1,2\}$. The conjecture concerns arbitrary finite nonempty tile sets and leaves the required threshold for sufficiently large $n$ unspecified.