Hello Brickboard!

I recently purchase (my dream car) a Volvo 240 GL Estate which was running when bought (a couple of weeks ago). Like all things I buy, it seems to have stopped working this week, so whilst I plan on posting another "Let the adventure begin" post to say hello to the forum, I figured getting the car running again should probably come first.

A bit of background -
- The engine is version 19 E.
- Car started (mostly) first time every time when purchased
- Car shuttered a bit when the engine was shut off
- The battery (which looks new) mysteriously died about 2 weeks after purchase despite relatively frequent drives
- I recharged the battery, the car started again (though something is continuing to drain it).
- Eventually the car failed to start
- The only mod I have made was installing a new single DIN car stereo (replacing a broken one that was in there when purchased).
- Describing myself as a novice mechanic would be an understatement (but everyone has to start somewhere, I'm keen to learn)

What I've tried to troubleshoot:
- Had a look at the fuse box, all fuses look ok. Have purchased new fuses to replace just in case. There is no visible corrosion on any of the fuses or connections.
- A mechanic came by and sprayed Easy Start into the engine from the air intake and it started - said that I had a fuel issue.

I've looked around the internet and seem numbers of things pointing to the ECU, the Fuel pump (actual and in-tank) as well as the Fuel Pump relay, but before I dove in head first and started pulling the car apart, I thought it wise to post and see if anyone has any notes on how to go about best deducing what the issue might be.

Also to note, there is a hose coming off the Intake that is currently not connected to anything (which seemed odd to me). It's highlighted in the image of my engine bay in blue - perhaps a good place to start?

Any help would be greatly appreciated!

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1989 Volvo 240 GL Estate, 19 E