Is EFTA membership compatible with EU Customs Union membership?

Question

Suppose a country wishes to join EFTA and also join the EU Customs Union (but not the EU itself), could it do so in the long term?

I suspect that membership of both is impossible because membership of the EU Customs Union means that the EU's tariffs will apply to trade with the other EFTA members. Is this right or is there maybe another reason why or why not?

Just to pre-empt misunderstanding...

1. I'm not asking about EEA or Single Market membership.
2. I am only interested in long term indefinite arrangements. Temporary membership of both organisations may or may not be possible (and the possibilities might be an interesting diversion) but that's not what I'm asking about.
3. A post-Brexit United Kingdom could be an example of a country that might want to join both organisations, but I'm asking for the general rule if any hypothetical country wanted to do this.

Answer 1

Quoth https://www.efta.int/About-EFTA/Frequently-asked-questions-EFTA-EEA-EFTA-membership-and-Brexit-328676 (My emphasis)

If the UK remains in a customs union with the EU, could it still join EFTA?

Art. 56.3 of the EFTA Convention states that a new EFTA member state ‘shall apply to become a party to the free trade agreements between the Member States on the one hand and third states, unions of states or international organisations on the other.’ As a member of a customs union, a country acceding to EFTA could not comply with this obligation. EFTA membership does not preclude from entering into a customs arrangement with the EU; existing EFTA countries govern their relation to the EU through different instruments.

(Note, this particular page somehow evaded my searching before I posted this question. My apologies.)