The exact overlap conjecture for self-similar sets and measures

Let $\Phi$ be an iterated function system and let $(p_i)_{i\in I}$ be a probability vector. For a finite word $\mathbf{a}=(a_1,\ldots,a_n)\in I^n$, write $\phi_{\mathbf{a}}=\phi_{a_1}\circ\cdots\circ\phi_{a_n}$. The IFS contains an exact overlap if there are distinct words $\mathbf{a},\mathbf{b}\in I^n$ with $\phi_{\mathbf{a}}=\phi_{\mathbf{b}}$. The associated self-similar set and measure have expected set dimension, expected measure dimension, and $L^q$ dimension bounds as described in the surrounding discussion. Exact overlap conjecture. If there is strict inequality in one of the expected set dimension, expected measure dimension, or $L^q$ dimension bounds, then $\Phi$ contains an exact overlap. This conjecture asserts that exact overlaps are the only mechanism producing strict dimension inequalities; it is described as an important folklore conjecture in fractal geometry, and no resolution is supplied here.