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When I search on the Internet for "spanish conditional progressive", I get "conditional of estar + present participle", but my Spanish teacher said to use the conditional form of the verb instead, as the conditional progressive.
Which is the right way to say "would be + -ing"; for example "I would be wearing"?
Both estaría llevando and llevaría are perfectly fine translations for I would be wearing
The issue is English strictly distinguishes the progressive from the simple aspect. For example you use simple present in English (What do drink?) you generally also need to in Spanish (¿Qué bebes?), but where you use present progressive in English (What are you drinking?) the use of present progressive is generally optional (¿Qué bebes/estás bebiendo?), depending on the level of focus you want to place on the action being ongoing and other factors. The same rule applies in other tenses.
As a general rule I tell my beginning students to just use progressive if they would in English — because it's rarely if ever wrong to use if English would use it, it just might not sound quite as natural. Over time you get a feel for understanding when it will sound most natural. What's probably happened is that your teacher feels tht given the context, it's sounds better to use simple conditional. That or they may be trying to steer you away from using overly complex tenses with multiple auxiliar verbs that Spanish tends to eschew.
This answer is supplementary to the one @guifa wrote.
I agree that the two forms are equivalent in Spanish 99% of the time, and I agree that your teacher might be training you towards a natural choice. But I can think of a tiny difference between the two forms.
A. Let's say you're co-writing a skit with a friend. You're brainstorming. Your friend outlines a possible scene. This is still a very tentative idea. Exploring the idea, you ask, "[If we had the scene start with you crashing a party for a top level UK official, like you said,] what would you be wearing to show your Greenpeace affiliation?" Here the verb form that would fit best would be, "¿Qué estarías llevando para demostrar...?"
B. Now let's imagine that your sister is overhauling her wardrobe with your help. There's some discussion about maybe getting rid of some formal wear taking up space in the closet. You say, "If Aunt Olivia set her wedding date for August, what would you wear?" This would be, "Si Tía Olivia se casara en agosto, ¿qué llevarías?"
