If Hajj has already become obligatory on a person, a general intention to perform Hajj fulfills their obligatory Hajj in this example. The condition that Hajj is already obligatory must stay with this ruling.
The measure-like aspect of Hajj time helps explain this, just as a general fasting intention counted toward Ramadan in the example already studied.
The explanation also points to the greater reward of the obligation and the severe consequence of neglecting it: a believer bearing Hajj's hardship would be understood to fulfill the obligation on them when the intention is general.
A general intention is not an explicit intention to perform only voluntary Hajj. That difference matters.
- A General Hajj Intention