Product of repdigits with consecutive lengths in the Fibonacci sequence.

From the well-known Fibonacci sequence, the number $$F_{10}=55=5\cdot 11$$ is an example not only as a repdigit (a number with only one distinct digit) but also as a product of two repdigits with consecutive lengths, 5 and 11. Here we find all the Fibonacci numbers that can be written as the product of k repdigits with consecutive lengths.