Tribonacci Exponents!

The following powers of 2 consist of all even digits:

$\begin{array}{lclcr} & & 2^{\color{#D61F06}1} &= &2\\ & & 2^{\color{#D61F06}2} &= & 4\\ 2^{\color{#D61F06}3} &= &2^{\color{#D61F06}{1+2}} &= & 8\\ 2^{\color{#D61F06}6} &= &2^{\color{#D61F06}{1 + 2 + 3}} &= & 64\\ 2^{\color{#D61F06}11} &= &2^{\color{#D61F06}{2 + 3 + 6}} &= & 2048 \end{array}$

Does $2^{\color{#D61F06}3+6+11}$ contain all even digits?

Generalization proofs are more than welcome here.

×