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The phrase "are sunk" is correct and usable in written English.
It can be used to describe a state of being submerged or overwhelmed, often in a metaphorical sense.
Example: "After the unexpected expenses, our savings are sunk, and we need to find a way to recover."
Alternatives: "are lost" or "are gone".
Dictionary
are sunk
verb
Past participle of sink
Exact(59)
All those costs are sunk.
They are sunk in their sockets.
Even devoted Atlanticists are sunk in gloom.
Now they are sunk in excessive gloom.
Some of the costs are sunk.
Some are sunk by their lack of ambition.
"I would think as many careers are sunk than launched".
A common assumption is that both are sunk.
Their capital costs are sunk and their marginal running costs are not high.
The cages are sunk six inches into the ground, but are partly visible among the grasses.
The sexual organs are sunk into the upper surface of the thallus.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com