For this premise to be true, the Holy Spirit must be a 'person' - a la trinity. Let's reiterate what we know with absolute confidence because the bible tells us - absolutely.
We're not told it is a 'person'.
We are told it comes from the Father and now the son. Notice it comes from the Father - not GOD!
...Spirit of truth which proceeds from the Father John 15:26
Most verses refer to 'it' in the Greek - not 'he' or 'him' as the translators persistently express.
Paul never refers to the Spirit in any greeting, The Holy Spirit has no name.
(Rev 14:1 ‘And I looked, the Lamb was standing on Mt Zion, and with him 144,000 having his name and the name of the Father on their foreheads’)
There is no biblical teaching that the premise of the Spirit being other than the Father is valid - it is an extra-biblical construct.
SO no, the Holy Spirit cannot 'taste death', it is the presence, power and expression of God and now the son, Jesus, who has risen to immortality - which he did not have before.
For Christ also died for sins once for all... having been put to death in the flesh, but made alive in the spirit 1 Pet 3:18
When Jesus said he will send the 'comforter', he explained further on that he and the Father will make their abode with us. They do this in and through the Spirit OF God. John 14:23
Luke 10:22 "All things have been committed to me by my Father. No one
knows who the Son is except the Father, and no one knows who the
Father is except the Son and those to whom the Son chooses to reveal
Again, the Holy Spirit is left out as if it doesn't exist in the centrality of who the Father is and who Jesus is. Of course it DOES exist...
God, the Father, is spirit, He is holy, he is Holy Spirit. Jesus also IS the Spirit 2 Cor 3:17
The subject of the passage is God - the Father. Jesus intersects the passage as he is obviously involved in what God is accomplishing. God remains the subject till the end of v16