In spite of what elementary textbooks tend to say, i.e., that the values of Comp are that (or 'null that'), whether, if and for (or 'null' for) in English, there is well established syntactic evidence that that initial analysis - dating back to the late 60's and early 70's - was incorrect.
As a matter of fact, whether is, like when, why, where, how much, what book, under what conditions, etc., a wh-phrase landing in Spec Focus, not a Comp head.
The simplest argument in this respect is that whether can be coordinated with other wh-phrases that must themselves occupy Spec Focus, whereas if cannot. Thus, in I would like to know whether and (,if so,) under what conditions I could postpone payment of my teaching fees, whether is clearly coordinated with a PP phrase and both land first in Spec Focus and then in Spec C (not in Focus and C!). Or, to take a different example, whether can also be coordinated with not, as in I would like to know whether or not I am admitted to the AI programme, where not is independently known to be a phrase base-generated in Spec Polarity (but not in the head of Polarity, as initially assumed). On the contrary, if, a C head, cannot be so coordinated, cf. * I would like to know if or not I am admitted to the AI programme.
Confirming evidence is available if you consider the etymology of whether (from OE hw-aeither) - or, for that matter, that of not (from n(e)+a+wiht, approximately 'not a bit', a negative noun phrase). Significantly, in Old and Middle English, hwaether did not interfere with the ascension of the finite verb into the head of FocusP. Although it is true that the finite verb did not have to ascend in hwaether-introduced clauses, and did not usually do (as in Modern English), the OED still cites quite a few OE and ME examples in which it does, yielding the word order ...hwaether + Verb + Subject...).
Of course, if hwaether had been a C (or Focus) head, such verb-raising would not have been possible in any case, because the finite verb is itself a head and can raise only into an empty head position (= Head Movement). Consequently, hwaether was already a specifier, as it is today, in Old and Middle English.