In defending His actions on the Sabbath, does Jesus use φιλέω rather than ἀγαπάω to imply equality with His Father?

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In defending His actions of healing the invalid and instructing him to pick up his bed and leave the pool, Jesus compares His actions to His Father’s and makes this statement:

For the Father loves (φιλεῖ) the Son and shows him all that he himself is doing. And greater works than these will he show him, so that you may marvel. (John 5:20 ESV)

Rather than agape love, Jesus says the Father’s love for Him is phileō. That is, in contrast to God’s love for the world which is ἠγάπησεν, the Father’s love for the Son is φιλεῖ.

Since phileō is used for kiss, there is a sense of proximity which is not necessarily present in agape. The BDAG notes "a more intimate relationship may be implied; one may extend ‘love’ in general, but close friendship is limited."¹One might even conclude the attempt to kill Jesus was exacerbated by for the Father loves/kisses the Son…

In this passage does Jesus use φιλέω to imply equality with His Father which would not be present if He had used ἀγαπάω?

_{1. Fredrick William Danker, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, The University Chicago Press, 2000, p. 1056}