See axiom (A5) in Section 2.4:

(A5) We can ask that the outer level validates the equality reflection rule, i.e. forms a model of extensional type theory. This is the case in all the example models we are interested in.

Equality reflection is supported in presheaf models, which justify conservativity over HoTT. The main problem with equality reflection is syntactical, in that we don't have good proof assistant support for it...

On Tue, 7 Jan 2020 at 23:11, Kristina Sojakova <sojakova.kristina@gmail.com> wrote:

Hello Rafael,

Thank you for the reference. I browsed the paper; it seems to me that
the theory does not appear to support identity reflection. I am looking
for a truly extensional form of equality (in addition to the usual one),
where equal terms are syntactically identified.

Kristina

On 1/7/2020 5:03 PM, Rafaël Bocquet wrote:
> Hello,
>
> I think that the paper "Two-Level Type Theory and Applications"
> (https://arxiv.org/abs/1705.03307), whose last version has been
> submitted on arXiv last month, answers these questions. One of the
> intended models of 2LTT is the presheaf category Ĉ over any model C of
> HoTT, and this presheaf model is conservative over C, essentially
> because the Yoneda embedding is fully faithful. This means that we can
> always work in 2LTT instead of HoTT.
>
> Rafaël
>
> On 1/7/20 8:59 PM, Kristina Sojakova wrote:
>> Dear all,
>>
>> I have been increasingly running into situations where I wished I had
>> an extensional equality type with a reflection rule in HoTT, in
>> addition to the intensional one to which univalence pertains. I know
>> that type systems with two equalities have been studied in the HoTT
>> community (e.g., VV's HTS), but last time I discussed this with
>> people it seemed the situation was not yet well-understood. So my
>> question is, what exactly goes wrong if we endow HoTT with an
>> extensional type?
>>
>> Thank you,
>>
>> Kristina
>>

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