meaning of the at once in a sentence

Solution 1:

My judgement is that the sentence is ungrammatical.

• ' ... seem at once X and Y' is short for ' ... seem at the same time to be X and Y' or more commonly ' ... seem to be both X and Y'.

For example

• 'The foxes seem at once curious and wary'

is a (very formal, perhaps literary) shorter paraphrase of

• 'The foxes seem curious and/but at the same time they seem wary'.

Note that this is a metaphorical usage of 'at once' / 'at the same time', unlike the literal ('at the same instant') usage in 'computers can perform many tasks at once'. These phrases may be substituted by 'both' (or 'not only ... but also'):

• 'The foxes seem both curious and wary'

But 'seem in which intelligibility can be put into more or less intelligible question' is unacceptable.

I'd amend to

But it is the linguistic arts that seem hospitable to the notion of intelligibility; further, it would seem / it is a fact that this intelligibility can be put into more or less intelligible question.

'To put N into question' for 'to investigate' (perhaps?) is highfalutin' too.