Are there Integers which are Sum of 2 Fifth Powers in 2 Ways?

Open Question
It is not known whether there exists an integer which equals the sum of two fifth powers in $2$ different ways.

That is, it is not known whether the Diophantine equation:
 * $a^5 + b^5 = c^5 + d^5$

has a solution.

Progress
It is known that no such solutions exist for sums up to $1 \cdotp 02 \times 10^{26}$.