Linear obstruction-size conjecture for graphs of bounded lettericity

Let $k$ be a positive integer, and call a graph an obstruction for the class of $k$-letter graphs if it is not a $k$-letter graph while all of its proper induced subgraphs are $k$-letter graphs. Linear obstruction-size conjecture. There is a constant $C>0$ such that every obstruction for the class of $k$-letter graphs has at most

$$C\cdot k$$

vertices. The conjecture would establish a linear bound on the size of minimal obstructions to bounded lettericity, improving the exponential bound obtained in the paper; the observed standard examples all have size within a small constant factor of their lettericity, but the conjecture remains open.