It's fine.

Evidently the contact material is suitable for higher inrush currents while being less susceptible to welding: https://www.electronicspecifier.com/products/power/relay-design-evolves-to-address-new-applications

Though I'd like a better reference than this. (If nothing else, you can contact Omron directly and see what they have to say. Or check if they have references regarding this.)

Relays don't specify inrush, or any time-current curves or anything like that, unfortunately. But they do assume typical -- quite nasty -- mains circuit conditions, including closing on capacitive loads or motor inrush, and inductive turn-off arcing. 5A is only the nominal load current, which you're well within.

Evidently the contact material is suitable for higher inrush currents while being less susceptible to welding: https://www.electronicspecifier.com/products/power/relay-design-evolves-to-address-new-applications

Though I'd like a better reference than this. (If nothing else, you can contact Omron directly and see what they have to say. Or check if they have references regarding this.) [Edit: the OP did contact them, but did not quote their response, or alternative suggestions.]

Relays don't specify inrush, or any time-current curves or anything like that, unfortunately. But they may assume typical -- quite nasty -- mains circuit conditions, including closing on capacitive loads or motor inrush, and inductive turn-off arcing. 5A is only the nominal load current, which you're well within.

Edit: from https://www.fa.omron.com.cn/upload/doc/qa/20080708084507281.pdf we have,

2. OMRON has also prepared the above [G2R series] relays with AgSnIn contacts, which are more tolerant of large inrush currents and physical movement compared with relays with standard contacts. When ordering, add "-ASI" to the model number. Example: G2R-1A-ASI
3. Standard, NO contact type relays are TV-3 class products in accordance with the TV standards of the UL/CSA. Models with AgSnIn contacts are TV-5 class products. Example: G2R-1A-ASI

No definition of what "TV-" ratings are.

It seems it's sometimes called a "tungsten" rating, because incandescent lamps have considerable inrush:
https://components.omron.com/us-en/products/basic-knowledge/relays/applications

I also see a reference suggesting they were developed for literal TVs (televisions), which used direct rectified mains power for a long time (i.e., hot chassis, mains rectified into large capacitor, huge inrush capacity), so inrush would be a big problem. I don't know that a lot of TVs were relay-operated though. Maybe relevant to early remote-controlled units (e.g. IR or ultrasonic activated relays)?

Possibly, VDE0435, UL508, CSA22.2, VDE0700, VDE0110, IEC 61810, etc. are the relevant standards. None of which are searchable on the internet at present so I'm afraid you'll have to consult someone with access to them to determine from where this rating and terminology actually comes from, and what precisely it entails.

In any case, the type G6R does not specify a "TV" rating, so likely does not handle exceptional inrush; but what exactly the baseline value is, is not apparent without access to these standards.

In absence of exact standards to draw from -- and in absence of a requirement for UL rating of the finished project -- I would be more than inclined to simply use it as-is.

If you must, additional inrush protection can be added, but what size of NTC is required, and what value is acceptable given the supply's ratings, isn't apparent from the datasheet. Replacing the relay with a higher rated one is the more reliable option.