Your sentence is fine. Whether can introduce an enumeration. Note that the binary use of whether is limited to the cases where the two alternatives are exhaustive (that is, they are the only two possible alternatives), sometimes even mutually exclusive. Guinlist explains
It is notable that the two possibilities combined by whether… or… are mutually exclusive opposites.
- Whether they WON or LOST, the children received a prize.
There is no other possibility than winning and losing.
However, when the alternatives are not exhaustive, they can constitute an enumeration headed by whether like in this structure: Whether + subject + verb followed by object or complement. Here is a very clear example:
Whether it [emotional abuse] is done by constant berating and belittling, by intimidation, or under the guise of “guidance,” “teaching,” or “advice,” the results are similar. (counselingcenter.uk)
Here the verb after whether is passive and the enumeration that follows is the agent complement of the verb.
In Thien's China, the thing that should have arrived is always still to come, whether it involves a missing person, a secret message or a better future. (The Guardian - Books)
Here the verb after whether is active, and the enumeration constitutes the direct object of the verb.
What qualifications were necessary in those who are to be created new lords: whether the humour of the prince, a sum of money to a court lady, or a design of strengthening a party opposite to the public interest, ever happened to be the motive in those advancements? (Gulliver's Travels, Jonathan Swift)
In such cases, the enumeration can include many possibilities. There is no rule that limits it to two. Though they give binary examples, dictionaries like M-W, Macmillan and OxfordL, all speak of two or more alternatives.
CAGEL does not explain this about whether, but on p. 990 you'll find this example:
The United Nations may not interfere in the political affairs of any nation, whether to unify it, federalise it, or balkanise it.
There is another one on p. 994:
Our thanks are due to all our staff, whether they be in the offices, the warehouses, or branches, for their help during this difficult time.
On p. 1001, it interprets this example:
- It meets with continuing hostility from those who see themselves as fostering and guarding serious art, [ whether it be in the theatre, in fiction, or on television].
Here we have a closed interrogative expressing an alternative question: the meaning is approximately "irrespective of the answer to the question 'Is it in the theatre, in fiction, or
on television?'':