Hope to get this conversation going

[Gray]

Hope to get this conversation going.

What is wrong with it? I could find only one result when I googled it. Don't we use this often?
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First lesson - English, not english. I, not i. ~A student of English

[Mister Micawber]

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Just too long a phrase to get good numbers of hits, I think. Seems OK to me.
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Native English teacher at Mister Micawber's

[Cerberus™]

Although its grammar seems fine, I wonder in what situation you would use this phrase? The word "this" probably indicates that the the person saying the phrase is actually in this conversation himself. But if he is talking about getting it going, it has not even started yet. And he probably would not say it as part of the pre-conversation anyway, because that would very awkward. That's why it may be an uncommon phrase as it is.
It gets nearly impossible if "hope" stands for "I hope".

Perhaps you could use this phrase when writing a story: "Peter almost bumped into Mary in the refectory. He made an attempt at conversation, but he was so nervous he kept stumbling over his words. Nevertheless, he still had hope that she would like him. Hope to get this conversation going. To impress her with his wit and knowledge. Surely she must like him then!"

[Gray]

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That sentence popped up when I was concluding a message and I wanted to convey that I would like to continue that (written) conversation. So, what do you think is appropriate there?

Even now, can I use -- hope to get the conversation (this one) going? (at this place?)
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First lesson - English, not english. I, not i. ~A student of English

[Cerberus™]

[Edit: I'm sorry, I hadn't read your reply yet when I wrote this.]

Or perhaps: "I run into my professor at the gym. We start talking. I try to gain his interest, hope to get this conversation going. But he seems rather in a hurry, and says he'd like to talk to me another time. I'll see him again in class anyway, so it doesn't really matter."
A story written like a diary, where the protagonist's thoughts are given in the present tense and from the first-person perspective.

[Cerberus™]

Hoping to get the conversation going implies that it might take some effort to start it off, that's why I wouldn't say it to someone.
Why not say "continue", as came to you naturally when you just described it? "I hope we may continue our conversation" would seem a better, more natural choice. Or you could just leave it out and let the message speak for itself: if you wrote questions or stimulating remarks, the other person will surely continue the conversation.

[Gray]

Oh, that is really interesting -- being concise and still implying the continuity in the conversation.

By the way, what Cerberus is?
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First lesson - English, not english. I, not i. ~A student of English

[Cerberus™]

Cerberus is the three-headed giant hound who guards the exit of the underworld against escaping souls in ancient Greek mythology. Cerberus is the Latinized form of Greek /Kerberos/.

[Gray]

[Cerberus™]

Ah, yes! I did not even know that, but I think you are right.