There may have been as many as 5 laryngeal sounds in PIE, one with no colouring effects (i.e. h₁), two a-colouring (normally mapped to h₂) and two o-colouring (mapped to h₃).
My reconstruction is as follows:
h₁ = /ʔ/ ( in syllabic positions);
h₂ = /x/ and /ɣ/;
h₃ = /χ(ʷ)/ and /ʁ̰(ʷ)/ (both may have been rounded to a degree, but this was not contrastive).
The syllabic laryngeals may have been pronounced with a faint schwa after the larygneal, which would have been itself subject to coloring; thus, /ɦə/=>/ə/, /xə/=>/xɐ/, /ɣə/=>/ɐ/, /χ(ʷ)ə/=>/χɵ/, /ʁ̰(ʷ)ə/=>/ʁ̰ɵ/.
The reason I reconstruct two a-colouring and two o-colouring laryngeals is to account for:
a) The appearance of a /h/ word initially in some Albanian words (such as herdhe);
b) The disappearance of laryngeals in Anatolian where it should not be expected (such as 𒀀𒀊𒉺);
c) The lack of aspiration of consonants preceding laryngeals where it would be expected in Indo-Iranian (compare पितृ (pitṛ) with no aspiration to स्थित (sthita));
d) The occasional voicing of consonants adjacent to an o-colouring laryngeal (*píph₃eti => *píbeti).
Thus, according to my reconstructions:
h₃rḗǵs deywós kʷe h₁ést nu pér h₂yu kʷe h₃rḗǵs n̥ h₂nepot Hyós h₁e suHnúm bʰuh₂ yéh₁t ǵʰewtréy h₁és Hyṓy h₃rḗǵs prḗḱst
Once there was a king. He was childless. The king wanted a son. He asked his priest: "May a son be born to me!" The priest said to the king: "Pray to the god Werunos." The king approached the god Werunos to pray now to the god. "Hear me, father Werunos!" The god Werunos came down from heaven. "What do you want?" "I want a son." "Let this be so," said the bright god Werunos. The king's lady bore a son.