Tribonacci Exponents!

The following powers of 2 consist of all even digits:

\[ \begin{array}{lclcr} & & 2^{\color{red}1} &= &2\\ & & 2^{\color{red}2} &= & 4\\ 2^{\color{red}3} &= &2^{\color{red}{1+2}} &= & 8\\ 2^{\color{red}6} &= &2^{\color{red}{1 + 2 + 3}} &= & 64\\ 2^{\color{red}11} &= &2^{\color{red}{2 + 3 + 6}} &= & 2048 \end{array} \]

Does \(2^{\color{red}3+6+11}\) contain all even digits?


Generalization proofs are more than welcome here.

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