Savannah Guthrie Visits Memorial for Her Missing Mother Nancy, on Video

12:52 PM PT -- Savannah Guthrie posted a still shot of the memorial scene on her Instagram Story just now … showing bright flowers against the greenery and blue sky in the background, writing how she and her family are feeling the love and prayers from well-wishers everywhere, adding … "please don't stop praying and hoping with us. bring her home."

Savannah Guthrie and family members were filmed outside their mother Nancy's Tucson, Arizona home this morning ... taking in a growing memorial for their mom one month into her search.

Watch the clip ... Savannah, her sister Annie, and brother-in-law Tommaso embraced as they laid flowers for Nancy. The memorial consisted of many bouquets and also signs with loving messages, along with what appears to be Nancy's "Missing" poster.

As we reported ... the still photo of the alleged masked assailant turned up no leads so far and the investigation into Nancy's kidnapping is becoming a "cold case."

However, there is still hope. A source told us they believe someone involved in the kidnapping will brag about it or insinuate their involvement 4 or 5 months down the line ... or someone who knows the identity of the kidnappers will either talk about it or go to authorities. This, the source said, is how they solve many of these crimes.