Why can’t lightning tx’s be signed with sighash single to allow for fee updates?

Why can’t lightning tx’s be signed with sighash single to allow for fee updates?

Are out-of-band fee payments to confirm a lightning closing transaction the only way to manage a situation where the sat/kw of a previously signed commitment tx are simply below market?

If additional inputs/outputs could be added, fees could be easily adjusted to the fee rate during channel closing and not at the time when the tx was signed...

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