Amdeberhan's Fibonacci conjecture for core partitions into distinct parts

Let an $(s,t)$-core partition be a partition that is simultaneously $s$-core and $t$-core, and let $F_n$ denote the $n$th Fibonacci number. Amdeberhan's conjecture. The number of $(s,s+1)$-core partitions into distinct parts equals

$$F_{s+1}.$$

This refines Anderson's Catalan enumeration of all $(s,s+1)$-core partitions; the conjecture concerns the restricted class of partitions into distinct parts and motivates the results proved in the paper.