Here is a video file my Xbox 360 refuses to play:

$ MP4Box -info video.mp4
* Movie Info *
        Timescale 90000 - Duration 02:18:33.365
        Fragmented File no - 2 track(s)
        File Brand mp42 - version 0
        Created: GMT Sat Jul 21 07:08:55 2012

File has root IOD (9 bytes)
Scene PL 0xff - Graphics PL 0xff - OD PL 0xff
Visual PL: ISO Reserved Profile (0x7f)
Audio PL: High Quality Audio Profile @ Level 2 (0x0f)
No streams included in root OD

iTunes Info:
        Encoder Software: HandBrake 0.9.6 2012022800

Track # 1 Info - TrackID 1 - TimeScale 90000 - Duration 02:18:33.235
Media Info: Language "Undetermined" - Type "vide:avc1" - 199318 samples
Visual Track layout: x=0 y=0 width=1280 height=688
MPEG-4 Config: Visual Stream - ObjectTypeIndication 0x21
AVC/H264 Video - Visual Size 1280 x 688
        AVC Info: 1 SPS - 1 PPS - Profile High @ Level 4.1
        NAL Unit length bits: 32
Self-synchronized

Track # 2 Info - TrackID 2 - TimeScale 48000 - Duration 02:18:33.365
Media Info: Language "English" - Type "soun:mp4a" - 389689 samples
MPEG-4 Config: Audio Stream - ObjectTypeIndication 0x40
MPEG-4 Audio MPEG-4 Audio AAC LC - 6 Channel(s) - SampleRate 48000
Synchronized on stream 1

$ avconv -i video.mp4
avconv version 0.8.4-4:0.8.4-0ubuntu0.12.04.1, Copyright (c) 2000-2012 the Libav     developers
  built on Nov  6 2012 16:51:33 with gcc 4.6.3
Input #0, mov,mp4,m4a,3gp,3g2,mj2, from 'video.mp4':
  Metadata:
    major_brand     : mp42
    minor_version   : 0
    compatible_brands: mp42isomavc1
    creation_time   : 2012-07-21 07:08:55
    encoder         : HandBrake 0.9.6 2012022800
  Duration: 02:18:33.36, start: 0.000000, bitrate: 2299 kb/s
    Stream #0.0(und): Video: h264 (High), yuv420p, 1280x688, 1973 kb/s, 23.98 fps, 90k tbr, 90k tbn, 180k tbc
    Metadata:
      creation_time   : 2012-07-21 07:08:55
    Stream #0.1(eng): Audio: aac, 48000 Hz, 5.1, s16, 319 kb/s
    Metadata:
      creation_time   : 2012-07-21 07:08:55
At least one output file must be specified

What tool, such as ffmpeg or mencoder, and what magic command line incantation should I use to transcode this file into a format Xbox 360 can play? I want the transcode process to retain as good video quality as possible.